{"id":1260,"date":"2021-12-02T19:37:08","date_gmt":"2021-12-03T00:37:08","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/4-1-solve-and-graph-linear-inequalities\/"},"modified":"2023-08-30T12:51:44","modified_gmt":"2023-08-30T16:51:44","slug":"solve-and-graph-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/solve-and-graph-linear-inequalities\/","title":{"raw":"4.1 Solve and Graph Linear Inequalities","rendered":"4.1 Solve and Graph Linear Inequalities"},"content":{"raw":"When given an equation, such as [latex]x = 4[\/latex] or [latex]x = -5,[\/latex] there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Symbol<\/th>\r\nMeaning<\/th>\r\n<\/tr>\r\n
\"Right<\/td>\r\n> Greater than<\/td>\r\n<\/tr>\r\n
\"Right<\/td>\r\n\u2264\u00a0Greater than or equal to<\/td>\r\n<\/tr>\r\n
\"Left<\/td>\r\n< Less than<\/td>\r\n<\/tr>\r\n
\"Left<\/td>\r\n\u2265\u00a0Less than or equal to<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n

Example 4.1.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nGiven a variable [latex]x[\/latex] such that [latex]x[\/latex] > [latex]4[\/latex], this means that [latex]x[\/latex] can be as close to 4 as possible but always larger. For [latex]x[\/latex] > [latex]4[\/latex], [latex]x[\/latex] can equal 5, 6, 7, 199. Even [latex]x =[\/latex] 4.000000000000001 is true, since [latex]x[\/latex] is larger than 4, so all of these are solutions to the inequality. The line graph of this inequality is shown below:\r\n\r\n\r\n\r\nWritten in interval notation, [latex]x[\/latex] > [latex]4[\/latex] is shown as [latex](4, \\infty)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nLikewise, if [latex]x < 3[\/latex], then [latex]x[\/latex] can be any value less than 3, such as 2, 1, \u2212102, even 2.99999999999. The line graph of this inequality is shown below:\r\n\r\n\r\n\r\nWritten in interval notation, [latex]x < 3[\/latex] is shown as [latex](-\\infty, 3)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFor greater than or equal (\u2265) and less than or equal (\u2264), the inequality starts at a defined number and then grows larger or smaller. For [latex]x \\ge 4,[\/latex] [latex]x[\/latex] can equal 5, 6, 7, 199, or 4. The line graph of this inequality is shown below:\r\n\r\n\r\n\r\nWritten in interval notation, [latex]x \\ge 4[\/latex] is shown as [latex][4, \\infty)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nIf [latex]x \\le 3[\/latex], then [latex]x[\/latex] can be any value less than or equal to 3, such as 2, 1, \u2212102, or 3. The line graph of this inequality is shown below:\r\n\r\n\r\n\r\nWritten in interval notation, [latex]x \\le 3[\/latex] is shown as [latex](-\\infty, 3].[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\nWhen solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:\r\n\r\nFirst, the sense flips when the inequality is divided or multiplied by a negative. For instance, in reducing [latex]-3x < 12[\/latex], it is necessary to divide both sides by \u22123. This leaves [latex]x[\/latex] > [latex]-4.[\/latex]\r\n\r\nSecond, the sense will flip over if the entire equation is flipped over. For instance, [latex]x[\/latex] > [latex]2[\/latex], when flipped over, would look like [latex]2 < x.[\/latex] In both cases, the 2 must be shown to be smaller than the [latex]x[\/latex], or the [latex]x[\/latex] is always greater than 2, no matter which side each term is on.\r\n
\r\n

Example 4.1.5<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSolve the inequality [latex]5-2x[\/latex] > [latex]11[\/latex] and show the solution on both a number line and in interval notation.\r\n\r\nFirst, subtract 5 from both sides:\r\n\r\n[latex]\\begin{array}{rrrrr}\r\n5&-&2x&\\ge &11 \\\\\r\n-5&&&&-5 \\\\\r\n\\hline\r\n&&-2x&\\ge &6\r\n\\end{array}[\/latex]\r\n\r\nDivide both sides by \u22122:\r\n\r\n[latex]\\begin{array}{rrr}\r\n\\dfrac{-2x}{-2} &\\ge &\\dfrac{6}{-2} \\\\\r\n\\end{array}[\/latex]\r\n\r\nSince the inequality is divided by a negative, it is necessary to flip the direction of the sense.\r\n\r\nThis leaves:\r\n\r\n[latex]x \\le -3[\/latex]\r\n\r\nIn interval notation, the solution is written as [latex](-\\infty, -3][\/latex].\r\n\r\nOn a number line, the solution looks like:\r\n\r\n\r\n\r\n<\/div>\r\n<\/div>\r\nInequalities can get as complex as the linear equations previously solved in this textbook. All the same patterns for solving inequalities are used for solving linear equations.\r\n
\r\n

Example 4.1.6<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSolve and give interval notation of [latex]3 (2x - 4) + 4x < 4 (3x - 7) + 8[\/latex].\r\n\r\nMultiply out the parentheses:\r\n\r\n[latex]6x - 12 + 4x < 12x - 28 + 8[\/latex]\r\n\r\nSimplify both sides:\r\n\r\n[latex]10x - 12 < 12x - 20[\/latex]\r\n\r\nCombine like terms:\r\n\r\n[latex]\\begin{array}{rrrrrrr}\r\n10x&-&12&<&12x&-&20 \\\\\r\n-12x&+&12&&-12x&+&12 \\\\\r\n\\hline\r\n&&-2x&<&-8&&\r\n\\end{array}[\/latex]\r\n\r\nThe last thing to do is to isolate [latex]x[\/latex] from the \u22122. This is done by dividing both sides by \u22122. Because both sides are divided by a negative, the direction of the sense must be flipped.\r\n\r\nThis means:\r\n\r\n[latex]\\dfrac{-2x}{-2}< \\dfrac{-8}{-2}[\/latex]\r\n\r\nWill end up looking like:\r\n\r\n\"x\r\n\r\nThe solution written on a number line is:\r\n\r\n\r\n\r\nWritten in interval notation, [latex]x[\/latex] > [latex]4[\/latex] is shown as [latex](4, \\infty)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\nFor questions 1 to 6, draw a graph for each inequality and give its interval notation.\r\n
    \r\n \t
  1. [latex]n[\/latex] > [latex]-5[\/latex]<\/li>\r\n \t
  2. [latex]n[\/latex] > [latex]4[\/latex]<\/li>\r\n \t
  3. [latex]-2 \\le k [\/latex]<\/li>\r\n \t
  4. [latex]1 \\ge k[\/latex]<\/li>\r\n \t
  5. [latex]5 \\ge x[\/latex]<\/li>\r\n \t
  6. [latex]-5 < x[\/latex]<\/li>\r\n<\/ol>\r\nFor questions 7 to 12, write the inequality represented on each number line and give its interval notation.\r\n
      \r\n \t
    1. <\/li>\r\n \t
    2. <\/li>\r\n \t
    3. <\/li>\r\n \t
    4. <\/li>\r\n \t
    5. <\/li>\r\n \t
    6. <\/li>\r\n<\/ol>\r\nFor questions 13 to 38, draw a graph for each inequality and give its interval notation.\r\n
        \r\n \t
      1. [latex]\\dfrac{x}{11}\\ge 10[\/latex]<\/li>\r\n \t
      2. [latex]-2 \\le \\dfrac{n}{13}[\/latex]<\/li>\r\n \t
      3. [latex]2 + r < 3[\/latex]<\/li>\r\n \t
      4. [latex]\\dfrac{m}{5} \\le -\\dfrac{6}{5}[\/latex]<\/li>\r\n \t
      5. [latex]8+\\dfrac{n}{3}\\ge 6[\/latex]<\/li>\r\n \t
      6. [latex]11[\/latex] > [latex]8+\\dfrac{x}{2}[\/latex]<\/li>\r\n \t
      7. [latex]2[\/latex] > [latex]\\dfrac{(a-2)}{5}[\/latex]<\/li>\r\n \t
      8. [latex]\\dfrac{(v-9)}{-4} \\le 2[\/latex]<\/li>\r\n \t
      9. [latex]-47 \\ge 8 -5x[\/latex]<\/li>\r\n \t
      10. [latex]\\dfrac{(6+x)}{12} \\le -1[\/latex]<\/li>\r\n \t
      11. [latex]-2(3+k) < -44[\/latex]<\/li>\r\n \t
      12. [latex]-7n-10 \\ge 60 [\/latex]<\/li>\r\n \t
      13. [latex]18 < -2(-8+p)[\/latex]<\/li>\r\n \t
      14. [latex]5 \\ge \\dfrac{x}{5} + 1[\/latex]<\/li>\r\n \t
      15. [latex]24 \\ge -6(m - 6)[\/latex]<\/li>\r\n \t
      16. [latex]-8(n - 5) \\ge 0[\/latex]<\/li>\r\n \t
      17. [latex]-r -5(r - 6) < -18[\/latex]<\/li>\r\n \t
      18. [latex]-60 \\ge -4( -6x - 3)[\/latex]<\/li>\r\n \t
      19. [latex]24 + 4b < 4(1 + 6b)[\/latex]<\/li>\r\n \t
      20. [latex]-8(2 - 2n) \\ge -16 + n[\/latex]<\/li>\r\n \t
      21. [latex]-5v - 5 < -5(4v + 1)[\/latex]<\/li>\r\n \t
      22. [latex]-36 + 6x[\/latex] > [latex]-8(x + 2) + 4x[\/latex]<\/li>\r\n \t
      23. [latex]4 + 2(a + 5) < -2( -a - 4)[\/latex]<\/li>\r\n \t
      24. [latex]3(n + 3) + 7(8 - 8n) < 5n + 5 + 2[\/latex]<\/li>\r\n \t
      25. [latex]-(k - 2)[\/latex] > [latex]-k - 20[\/latex]<\/li>\r\n \t
      26. [latex]-(4 - 5p) + 3 \\ge -2(8 - 5p)[\/latex]<\/li>\r\n<\/ol>\r\nAnswer Key 4.1<\/a>","rendered":"

        When given an equation, such as [latex]x = 4[\/latex] or [latex]x = -5,[\/latex] there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:<\/p>\n\n\n\n\n\n\n\n
        Symbol<\/th>\nMeaning<\/th>\n<\/tr>\n
        \"Right<\/td>\n> Greater than<\/td>\n<\/tr>\n
        \"Right<\/td>\n\u2264\u00a0Greater than or equal to<\/td>\n<\/tr>\n
        \"Left<\/td>\n< Less than<\/td>\n<\/tr>\n
        \"Left<\/td>\n\u2265\u00a0Less than or equal to<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        \n
        \n

        Example 4.1.1<\/p>\n<\/header>\n

        \n

        Given a variable [latex]x[\/latex] such that [latex]x[\/latex] > [latex]4[\/latex], this means that [latex]x[\/latex] can be as close to 4 as possible but always larger. For [latex]x[\/latex] > [latex]4[\/latex], [latex]x[\/latex] can equal 5, 6, 7, 199. Even [latex]x =[\/latex] 4.000000000000001 is true, since [latex]x[\/latex] is larger than 4, so all of these are solutions to the inequality. The line graph of this inequality is shown below:<\/p>\n

        \"image\"<\/p>\n

        Written in interval notation, [latex]x[\/latex] > [latex]4[\/latex] is shown as [latex](4, \\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        Example 4.1.2<\/p>\n<\/header>\n

        \n

        Likewise, if [latex]x < 3[\/latex], then [latex]x[\/latex] can be any value less than 3, such as 2, 1, \u2212102, even 2.99999999999. The line graph of this inequality is shown below:\n\n\"image\"<\/p>\n

        Written in interval notation, [latex]x < 3[\/latex] is shown as [latex](-\\infty, 3)[\/latex].\n\n<\/div>\n<\/div>\n

        \n
        \n

        Example 4.1.3<\/p>\n<\/header>\n

        \n

        For greater than or equal (\u2265) and less than or equal (\u2264), the inequality starts at a defined number and then grows larger or smaller. For [latex]x \\ge 4,[\/latex] [latex]x[\/latex] can equal 5, 6, 7, 199, or 4. The line graph of this inequality is shown below:<\/p>\n

        \"image\"<\/p>\n

        Written in interval notation, [latex]x \\ge 4[\/latex] is shown as [latex][4, \\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        Example 4.1.4<\/p>\n<\/header>\n

        \n

        If [latex]x \\le 3[\/latex], then [latex]x[\/latex] can be any value less than or equal to 3, such as 2, 1, \u2212102, or 3. The line graph of this inequality is shown below:<\/p>\n

        \"image\"<\/p>\n

        Written in interval notation, [latex]x \\le 3[\/latex] is shown as [latex](-\\infty, 3].[\/latex]<\/p>\n<\/div>\n<\/div>\n

        When solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:<\/p>\n

        First, the sense flips when the inequality is divided or multiplied by a negative. For instance, in reducing [latex]-3x < 12[\/latex], it is necessary to divide both sides by \u22123. This leaves [latex]x[\/latex] > [latex]-4.[\/latex]\n\nSecond, the sense will flip over if the entire equation is flipped over. For instance, [latex]x[\/latex] > [latex]2[\/latex], when flipped over, would look like [latex]2 < x.[\/latex] In both cases, the 2 must be shown to be smaller than the [latex]x[\/latex], or the [latex]x[\/latex] is always greater than 2, no matter which side each term is on.\n\n\n

        \n
        \n

        Example 4.1.5<\/p>\n<\/header>\n

        \n

        Solve the inequality [latex]5-2x[\/latex] > [latex]11[\/latex] and show the solution on both a number line and in interval notation.<\/p>\n

        First, subtract 5 from both sides:<\/p>\n

        [latex]\\begin{array}{rrrrr} 5&-&2x&\\ge &11 \\\\ -5&&&&-5 \\\\ \\hline &&-2x&\\ge &6 \\end{array}[\/latex]<\/p>\n

        Divide both sides by \u22122:<\/p>\n

        [latex]\\begin{array}{rrr} \\dfrac{-2x}{-2} &\\ge &\\dfrac{6}{-2} \\\\ \\end{array}[\/latex]<\/p>\n

        Since the inequality is divided by a negative, it is necessary to flip the direction of the sense.<\/p>\n

        This leaves:<\/p>\n

        [latex]x \\le -3[\/latex]<\/p>\n

        In interval notation, the solution is written as [latex](-\\infty, -3][\/latex].<\/p>\n

        On a number line, the solution looks like:<\/p>\n

        \"image\"<\/p>\n<\/div>\n<\/div>\n

        Inequalities can get as complex as the linear equations previously solved in this textbook. All the same patterns for solving inequalities are used for solving linear equations.<\/p>\n

        \n
        \n

        Example 4.1.6<\/p>\n<\/header>\n

        \n

        Solve and give interval notation of [latex]3 (2x - 4) + 4x < 4 (3x - 7) + 8[\/latex].\n\nMultiply out the parentheses:\n\n[latex]6x - 12 + 4x < 12x - 28 + 8[\/latex]\n\nSimplify both sides:\n\n[latex]10x - 12 < 12x - 20[\/latex]\n\nCombine like terms:\n\n[latex]\\begin{array}{rrrrrrr} 10x&-&12&<&12x&-&20 \\\\ -12x&+&12&&-12x&+&12 \\\\ \\hline &&-2x&<&-8&& \\end{array}[\/latex]\n\nThe last thing to do is to isolate [latex]x[\/latex] from the \u22122. This is done by dividing both sides by \u22122. Because both sides are divided by a negative, the direction of the sense must be flipped.\n\nThis means:\n\n[latex]\\dfrac{-2x}{-2}< \\dfrac{-8}{-2}[\/latex]\n\nWill end up looking like:\n\n\"x<\/p>\n

        The solution written on a number line is:<\/p>\n

        \"image\"<\/p>\n

        Written in interval notation, [latex]x[\/latex] > [latex]4[\/latex] is shown as [latex](4, \\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n

        Questions<\/h1>\n

        For questions 1 to 6, draw a graph for each inequality and give its interval notation.<\/p>\n

          \n
        1. [latex]n[\/latex] > [latex]-5[\/latex]<\/li>\n
        2. [latex]n[\/latex] > [latex]4[\/latex]<\/li>\n
        3. [latex]-2 \\le k[\/latex]<\/li>\n
        4. [latex]1 \\ge k[\/latex]<\/li>\n
        5. [latex]5 \\ge x[\/latex]<\/li>\n
        6. [latex]-5 < x[\/latex]<\/li>\n<\/ol>\n

          For questions 7 to 12, write the inequality represented on each number line and give its interval notation.<\/p>\n

            \n
          1. \"image\"<\/li>\n
          2. \"image\"<\/li>\n
          3. \"image\"<\/li>\n
          4. \"image\"<\/li>\n
          5. \"image\"<\/li>\n
          6. \"image\"<\/li>\n<\/ol>\n

            For questions 13 to 38, draw a graph for each inequality and give its interval notation.<\/p>\n

              \n
            1. [latex]\\dfrac{x}{11}\\ge 10[\/latex]<\/li>\n
            2. [latex]-2 \\le \\dfrac{n}{13}[\/latex]<\/li>\n
            3. [latex]2 + r < 3[\/latex]<\/li>\n
            4. [latex]\\dfrac{m}{5} \\le -\\dfrac{6}{5}[\/latex]<\/li>\n
            5. [latex]8+\\dfrac{n}{3}\\ge 6[\/latex]<\/li>\n
            6. [latex]11[\/latex] > [latex]8+\\dfrac{x}{2}[\/latex]<\/li>\n
            7. [latex]2[\/latex] > [latex]\\dfrac{(a-2)}{5}[\/latex]<\/li>\n
            8. [latex]\\dfrac{(v-9)}{-4} \\le 2[\/latex]<\/li>\n
            9. [latex]-47 \\ge 8 -5x[\/latex]<\/li>\n
            10. [latex]\\dfrac{(6+x)}{12} \\le -1[\/latex]<\/li>\n
            11. [latex]-2(3+k) < -44[\/latex]<\/li>\n
            12. [latex]-7n-10 \\ge 60[\/latex]<\/li>\n
            13. [latex]18 < -2(-8+p)[\/latex]<\/li>\n
            14. [latex]5 \\ge \\dfrac{x}{5} + 1[\/latex]<\/li>\n
            15. [latex]24 \\ge -6(m - 6)[\/latex]<\/li>\n
            16. [latex]-8(n - 5) \\ge 0[\/latex]<\/li>\n
            17. [latex]-r -5(r - 6) < -18[\/latex]<\/li>\n
            18. [latex]-60 \\ge -4( -6x - 3)[\/latex]<\/li>\n
            19. [latex]24 + 4b < 4(1 + 6b)[\/latex]<\/li>\n
            20. [latex]-8(2 - 2n) \\ge -16 + n[\/latex]<\/li>\n
            21. [latex]-5v - 5 < -5(4v + 1)[\/latex]<\/li>\n
            22. [latex]-36 + 6x[\/latex] > [latex]-8(x + 2) + 4x[\/latex]<\/li>\n
            23. [latex]4 + 2(a + 5) < -2( -a - 4)[\/latex]<\/li>\n
            24. [latex]3(n + 3) + 7(8 - 8n) < 5n + 5 + 2[\/latex]<\/li>\n
            25. [latex]-(k - 2)[\/latex] > [latex]-k - 20[\/latex]<\/li>\n
            26. [latex]-(4 - 5p) + 3 \\ge -2(8 - 5p)[\/latex]<\/li>\n<\/ol>\n

              Answer Key 4.1<\/a><\/p>\n","protected":false},"author":90,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1260","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1241,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1260","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1260\/revisions"}],"predecessor-version":[{"id":2090,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1260\/revisions\/2090"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1241"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1260\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1260"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1260"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1260"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}